Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [280,3,Mod(69,280)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(280, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("280.69");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 280 = 2^{3} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 280.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | no |
| Analytic conductor: | \(7.62944740209\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{14})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 49 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} + 49 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 7\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 7\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).
| \(n\) | \(57\) | \(71\) | \(141\) | \(241\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) | \(-1\) |
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 69.1 |
|
− | 2.00000i | − | 5.74166i | −4.00000 | −1.12917 | + | 4.87083i | −11.4833 | − | 7.00000i | 8.00000i | −23.9666 | 9.74166 | + | 2.25834i | |||||||||||||||||||||||
| 69.2 | − | 2.00000i | 1.74166i | −4.00000 | −4.87083 | + | 1.12917i | 3.48331 | − | 7.00000i | 8.00000i | 5.96663 | 2.25834 | + | 9.74166i | |||||||||||||||||||||||||
| 69.3 | 2.00000i | − | 1.74166i | −4.00000 | −4.87083 | − | 1.12917i | 3.48331 | 7.00000i | − | 8.00000i | 5.96663 | 2.25834 | − | 9.74166i | |||||||||||||||||||||||||
| 69.4 | 2.00000i | 5.74166i | −4.00000 | −1.12917 | − | 4.87083i | −11.4833 | 7.00000i | − | 8.00000i | −23.9666 | 9.74166 | − | 2.25834i | ||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 56.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-14}) \) |
| 5.b | even | 2 | 1 | inner |
| 280.c | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 280.3.c.e | ✓ | 4 |
| 4.b | odd | 2 | 1 | 1120.3.c.e | 4 | ||
| 5.b | even | 2 | 1 | inner | 280.3.c.e | ✓ | 4 |
| 7.b | odd | 2 | 1 | 280.3.c.f | yes | 4 | |
| 8.b | even | 2 | 1 | 280.3.c.f | yes | 4 | |
| 8.d | odd | 2 | 1 | 1120.3.c.f | 4 | ||
| 20.d | odd | 2 | 1 | 1120.3.c.e | 4 | ||
| 28.d | even | 2 | 1 | 1120.3.c.f | 4 | ||
| 35.c | odd | 2 | 1 | 280.3.c.f | yes | 4 | |
| 40.e | odd | 2 | 1 | 1120.3.c.f | 4 | ||
| 40.f | even | 2 | 1 | 280.3.c.f | yes | 4 | |
| 56.e | even | 2 | 1 | 1120.3.c.e | 4 | ||
| 56.h | odd | 2 | 1 | CM | 280.3.c.e | ✓ | 4 |
| 140.c | even | 2 | 1 | 1120.3.c.f | 4 | ||
| 280.c | odd | 2 | 1 | inner | 280.3.c.e | ✓ | 4 |
| 280.n | even | 2 | 1 | 1120.3.c.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 280.3.c.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 280.3.c.e | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 280.3.c.e | ✓ | 4 | 56.h | odd | 2 | 1 | CM |
| 280.3.c.e | ✓ | 4 | 280.c | odd | 2 | 1 | inner |
| 280.3.c.f | yes | 4 | 7.b | odd | 2 | 1 | |
| 280.3.c.f | yes | 4 | 8.b | even | 2 | 1 | |
| 280.3.c.f | yes | 4 | 35.c | odd | 2 | 1 | |
| 280.3.c.f | yes | 4 | 40.f | even | 2 | 1 | |
| 1120.3.c.e | 4 | 4.b | odd | 2 | 1 | ||
| 1120.3.c.e | 4 | 20.d | odd | 2 | 1 | ||
| 1120.3.c.e | 4 | 56.e | even | 2 | 1 | ||
| 1120.3.c.e | 4 | 280.n | even | 2 | 1 | ||
| 1120.3.c.f | 4 | 8.d | odd | 2 | 1 | ||
| 1120.3.c.f | 4 | 28.d | even | 2 | 1 | ||
| 1120.3.c.f | 4 | 40.e | odd | 2 | 1 | ||
| 1120.3.c.f | 4 | 140.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(280, [\chi])\):
|
\( T_{3}^{4} + 36T_{3}^{2} + 100 \)
|
|
\( T_{17} \)
|
|
\( T_{19}^{2} - 12T_{19} - 650 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{2} \)
|
| $3$ |
\( T^{4} + 36T^{2} + 100 \)
|
| $5$ |
\( T^{4} + 12 T^{3} + \cdots + 625 \)
|
| $7$ |
\( (T^{2} + 49)^{2} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + 676 T^{2} + 96100 \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( (T^{2} - 12 T - 650)^{2} \)
|
| $23$ |
\( (T^{2} + 2016)^{2} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( (T^{2} + 108 T - 1130)^{2} \)
|
| $61$ |
\( (T^{2} - 12 T - 7370)^{2} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( (T^{2} - 8064)^{2} \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( (T^{2} - 8064)^{2} \)
|
| $83$ |
\( T^{4} + 27556 T^{2} + 172396900 \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} \)
|
| show more | |
| show less |